Math 561 Spring 2020, Lecture notes of Probability and Statistics

Download Math 561 Spring 2020 and more Lecture notes Probability and Statistics in PDF only on Docsity!

Math 561 Spring 2020

Renming Song

University of Illinois at Urbana-Champaign

April 9, 2020

Outline

In this section,, we will give necessary and sufficient conditions for a martingale to converge in L^1. The key to this is the concept of uniform integrability.

Definition

A collection {Xi , i ∈ I} of rv’s is said to be uniformly integrable if

lim M↑∞

sup i∈I

E(|Xi |; |Xi | > M)

If {Xi , i ∈ I} is uniformly integrable, by picking M large enough so that

sup i∈I

E(|Xi |; |Xi | > M) ≤ 1 ,

we get sup i∈I

E(|Xi |) ≤ M + 1 < ∞.

So {Xi , i ∈ I} is bounded in L^1.

In this section,, we will give necessary and sufficient conditions for a martingale to converge in L^1. The key to this is the concept of uniform integrability.

Definition

A collection {Xi , i ∈ I} of rv’s is said to be uniformly integrable if

lim M↑∞

sup i∈I

E(|Xi |; |Xi | > M)

If {Xi , i ∈ I} is uniformly integrable, by picking M large enough so that

sup i∈I

E(|Xi |; |Xi | > M) ≤ 1 ,

we get sup i∈I

E(|Xi |) ≤ M + 1 < ∞.

So {Xi , i ∈ I} is bounded in L^1.

If there is Y ∈ L^1 such that |Xi | ≤ Y , i ∈ I, then {Xi , i ∈ I} is uniformly integrable.

If {Xi , i ∈ I} is bounded in Lp^ for some p > 1, then {Xi , i ∈ I} is uniformly integrable.

In fact, let a = supi∈I E[|Xi |p]. Then for any c > 0,

E[|Xi |; |Xi | > c] ≤

cp−^1 E[|Xi |p; |Xi | > c] ≤

E[|Xi |p] cp−^1

a cp−^1

thus {Xi , i ∈ I} is uniformly integrable.

If there is Y ∈ L^1 such that |Xi | ≤ Y , i ∈ I, then {Xi , i ∈ I} is uniformly integrable.

If {Xi , i ∈ I} is bounded in Lp^ for some p > 1, then {Xi , i ∈ I} is uniformly integrable.

In fact, let a = supi∈I E[|Xi |p]. Then for any c > 0,

E[|Xi |; |Xi | > c] ≤

cp−^1 E[|Xi |p; |Xi | > c] ≤

E[|Xi |p] cp−^1

a cp−^1

thus {Xi , i ∈ I} is uniformly integrable.

Theorem 5.5.

Given a probability space (Ω, F 0 , P) and an X ∈ L^1 , then {E(X |F) : F is a sub-σ-field of F 0 } is uniformly integrable.

Proof of Theorem 5.5.

Note that

P(E(|X ||F) > c) ≤

c

E[E(|X ||F)] =

E(|X |)

c

Hence, for any δ > 0,

E[E(|X ||F); E(|X ||F) > c] = E[|X |; E(|X ||F) > c] ≤ δP(E(|X ||F) > c) + E[|X |; |X | > δ]

≤

δ c E[|X |] + E[|X |; |X | > δ].

Theorem 5.5.

Given a probability space (Ω, F 0 , P) and an X ∈ L^1 , then {E(X |F) : F is a sub-σ-field of F 0 } is uniformly integrable.

Proof of Theorem 5.5.

Note that

P(E(|X ||F) > c) ≤

c

E[E(|X ||F)] =

E(|X |)

c

Hence, for any δ > 0,

E[E(|X ||F); E(|X ||F) > c] = E[|X |; E(|X ||F) > c] ≤ δP(E(|X ||F) > c) + E[|X |; |X | > δ]

≤

δ c E[|X |] + E[|X |; |X | > δ].

Proof of Theorem 5.5.1 (cont)

For any  > 0, choose δ > 0 so that E[|X |; |X | > δ] ≤ /2. Then for c ≥ 2 δE[|X |]/, we have

E[E(|X ||F); E(|X ||F) > c] ≤ .

Thus {E(|X ||F) : F is a sub-σ-field of F 0 } is uniformly integrable. From which the desired conclusion follows immediately.

Here is an equivalent characterization of uniform integrability.

Theorem

In order for {Xi , i ∈ I} to be uniformly integrable, it is necessary and sufficient that the following two conditions are satisfied:

(i) a = supi∈I E[|Xi |] < ∞; (ii) for any  > 0, there exists δ > 0 such that for any event A with P(A) ≤ δ, it holds that

E[|Xi |; A] ≤ .

Proof of sufficiency

Suppose that (i) and (ii) hold. For any  > 0, choose δ > 0 so that (ii) holds. Then for c ≥ a/δ, we have

P(|Xi | ≥ c) ≤

c

E[|Xi |] ≤

a c

≤ δ.

So by (ii), we have

E[|Xi |; |Xi | > c] ≤ , i ∈ I.

Theorem 5.5.

If Xn → X in probability, then the following are equivalent

(i) {Xn : n ≥ 0 } is uniformly integrable; (ii) Xn → X in L^1 ; (iii) E|Xn| → E|X | < ∞.

Proof of Theorem 5.5.

(i)⇒(ii) Let

ϕM (x) =

M, if x ≥ M x, if |x| ≤ M −M, if x ≤ −M.

The triangle inequality implies

|Xn − X | ≤ |Xn − ϕM (Xn)| + |ϕM (Xn) − ϕM (X )| + |ϕM (X ) − X |.

Proof of Theorem 5.5.2 (cont)

Since |ϕM (Y ) − Y | = (|Y | − M)+^ ≤ |Y | (^1) {|Y |≥M}, taking expectation gives

E|Xn − X | ≤ E|ϕM (Xn) − ϕM (X )| + E[|Xn|; |Xn| > M] + E[|X |; |X | > M].

Since ϕM (Xn) → ϕM (X ) in probability, the 1st term → 0 by the BCT. If  > 0 and M is large, uniform integrability implies the 2nd ≤ . For the 3rd term, we note that uniform integrability implies supn E|Xn| < ∞, so Fatou implies E|X | < ∞, by taking M larger if necessary, the 3rd term ≤ . Combining the 3 facts we get supn E|Xn − X | ≤ 2 . Since  is arbitrary, this implies (ii). (ii)⇒(iii) Jensen’s inequality implies

|E|Xn| − E|X || ≤ E‖Xn| − |X || ≤ E|Xn − X | → 0.

Proof of Theorem 5.5.2 (cont)

(iii)⇒(i) Let

ψM (x) =

x, on [ 0 , M − 1 ] 0 , on [M, ∞) linear , on [M − 1 , M].

The DCT implies that, if M is large, E|X | − EψM (|X |) ≤ /2. As in the 1st part of the proof, the BCT implies EψM (|Xn|) → EψM (|X |), so using (iii) we get that if n ≥ n 0 ,

E(|Xn| : |Xn| > M) ≤ E|Xn| − EψM (|Xn|) ≤ E|X | − EψM (|X |) +

By choosing M larger we can make E(|Xn| : |Xn| > M) ≤  for 0 ≤ n ≤ n 0 , so {Xn} is uniformly integrable.